New Perspectives on Polyhedral Molecules and their Crystal Structures Santiago Alvarez, Jorge Echeverria To cite this version: Santiago Alvarez, Jorge Echeverria. New Perspectives on Polyhedral Molecules and their Crystal Structures. Journal of Physical Organic Chemistry, Wiley, 2010, 23 (11), pp.1080. 10.1002/poc.1735. hal-00589441 HAL Id: hal-00589441 https://hal.archives-ouvertes.fr/hal-00589441 Submitted on 29 Apr 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Journal of Physical Organic Chemistry New Perspectives on Polyhedral Molecules and their Crystal Structures For Peer Review Journal: Journal of Physical Organic Chemistry Manuscript ID: POC-09-0305.R1 Wiley - Manuscript type: Research Article Date Submitted by the 06-Apr-2010 Author: Complete List of Authors: Alvarez, Santiago; Universitat de Barcelona, Departament de Quimica Inorganica Echeverria, Jorge; Universitat de Barcelona, Departament de Quimica Inorganica continuous shape measures, stereochemistry, shape maps, Keywords: polyhedranes http://mc.manuscriptcentral.com/poc Page 1 of 20 Journal of Physical Organic Chemistry 1 2 3 4 5 6 7 8 9 10 New Perspectives on Polyhedral Molecules and their Crystal Structures 11 12 Santiago Alvarez, Jorge Echeverría 13 14 15 Departament de Química Inorgànica and Institut de Química Teòrica i Computacional, 16 Universitat de Barcelona, Martí i Franquès 1-11, 08028 Barcelona (Spain). Fax: +34-93-490 17 18 7725; e-mail: [email protected] 19 20 For Peer Review 21 22 23 Abstract 24 25 26 Relevant families of ideal polyhedra (Platonic, Archimedean, prisms Johnson, and 27 Fullerenes) are briefly summarized, and an overview of polyhedral alkanes and alkenes, existing 28 29 or hypothetical, is presented. The assignment of a polyhedral shape to a specific compound with 30 the help of continuous shape measures and derived tools is also briefly discussed, and 31 32 application of shape analysis to cyclic molecules such as cyclobutane, cyclohexane and 33 cyclooctatetraene is presented to illustrate the usefulness of ideal polyhedra in the 34 35 stereochemical description of non-polyhedral molecules. Finally, the presence of latent 36 octahedral symmetry in icosahedral polyhedra is used to design new molecules with nested 37 38 shells of the two supposedly incompatible symmetries, and to explain the cubic crystal structures 39 40 of icosahedral molecules such as dodecahedrane and Buckminsterfullerene. 41 42 43 44 45 46 47 48 49 50 Keywords: Continuous shape measures, stereochemistry, polyhedral molecules, polyhedranes, 51 polyhedrenes. 52 53 54 55 56 57 58 59 60 http://mc.manuscriptcentral.com/poc Journal of Physical Organic Chemistry Page 2 of 20 1 2 3 4 5 I filled near full with Pease and Water, the iron Pot 6 and laid on the Pease a leaden cover [...] the Pease 7 dilated [... and] what they increased in bulk was [...] 8 9 pressed into the interstices of the Pease, which they 10 adequately filled up, being thereby formed into pretty 11 regular Dodecahedrons. 12 13 14 Stephen Hales, Vegetable Staticks, 1727. 15 16 17 Introduction 18 19 While the Platonic tetrahedral coordination around a carbon atom, proposed in 1874 by 20 For Peer Review 21 Jacobus Henricus Van 't Hoff and Joseph Achille Le Bel, represented the cornerstone of a new 22 discipline, stereochemistry, molecules with polyhedral skeletons in which a carbon atom 23 24 occupies each vertex, are relatively newcomers to the world of synthesized and well 25 characterized molecules. The oldest member of the family of polyhedranes, cubane, was 26 [1] 27 reported in 1964. A subsequent geometrical analysis of prisms, Platonic and Archimedean 28 polyhedra, pointed to those that could be made of sp3-hybridized carbon atoms.[2] Tetrahedrane 29 [3] 30 had to wait more than a decade to see the light, and soon after dodecahedrane could be 31 synthesized.[4] Much more recent is the discovery of buckminsterfullerene, of formula C , with 32 60 33 the shape of a truncated icosahedron.[5] Besides the Platonic solids, there are other sorts of 34 35 semiregular polyhedra that can be made as purely organic molecules. These include 36 Archimedean solids, prisms and some of the 92 Johnson polyhedra.[6] 37 38 39 With the structure of a polyhedral molecule at hand, we must address two relevant 40 stereochemical questions: Which ideal polyhedron represents best its stereochemistry? How 41 42 similar is the molecular structure to the ideal shape? Avnir and coworkers have proposed that 43 symmetry[7] and shape[8] should be treated as continuous properties and defined continuous 44 45 symmetry measures (CSM) and continuous shape measures (CShM). These parameters allow us 46 to calibrate the deviation of structures from a given symmetry or shape at the same scale, 47 48 independent of their size or number of vertices. Later on, we showed that one can define a 49 minimal distortion interconversion path between two polyhedra in terms of continuous shape 50 51 measures and therefore numerically evaluate not only the deviation of a given structure from a 52 53 particular polyhedron, but also its deviation from the minimal distortion path between two 54 reference polyhedra.[9] 55 56 57 Herein we wish to present an overview of the main polyhedral organic molecules. We will 58 also present examples of minimal distortion paths between a polygon and a polyhedron to 59 60 organize structural and conformational diversity of cyclohexyl and cyclooctatetraene derivatives. Finally, we will discuss the latency of cubic symmetry in icosahedral and 2 http://mc.manuscriptcentral.com/poc Page 3 of 20 Journal of Physical Organic Chemistry 1 2 3 4 tetrahedral molecules, and on the implications for crystal packing and for the design of metal- 5 organic frameworks. 6 7 8 Shape and Symmetry of Ideal Polyhedra 9 10 Although molecular shape and symmetry are intimately associated, it is important to stress 11 12 here the main differences between these two properties. Let us consider as an example the two 13 Archimedean polyhedra with 24 vertices shown in Figure 1, the truncated cube and the truncated 14 15 octahedron: they both have octahedral symmetry (i.e., they belong to the Oh symmetry point 16 17 group), but they differ in their number and type of faces. In other words, they have the same 18 symmetry but different shapes. 19 20 For Peer Review 21 In brief, we say that two objects (molecules) have the same shape if they differ only in size, 22 position or orientation in space. Alternatively, we can say that two objects (molecules) have the 23 24 same shape if they can be superimposed by combinations of translations, rotations and isotropic 25 scaling.[10] On the other hand, two objects have the same symmetry if they remain 26 27 indistinguishable after application of the same set of symmetry operations. It follows that two 28 objects with different shapes may have the same symmetry, and we can therefore conclude that 29 30 shape is a more stringent criterion than symmetry, as illustrated by the examples in Figure 1. 31 There are, however some cases in which shape and symmetry are equivalent, and these 32 33 correspond precisely to the Platonic polyhedra, since, e. g., all four vertex polyhedra with 34 35 tetrahedral symmetry have the same shape, and the same happens for the octahedron, the cube, 36 the dodecahedron and the icosahedron. Among the Archimedean polyhedra, only for the 37 38 cuboctahedron and the icosidodecahedron are shape and symmetry equivalent. 39 40 41 42 43 44 45 46 47 48 49 50 51 52 Figure 1. Two Archimedean polyhedra with octahedral symmetry and 24 vertices: the truncated 53 cube (left) and the truncated octahedron (right). 54 55 56 The set of Platonic solids (tetrahedron, octahedron, cube, icosahedron and dodecahedron) 57 are the most regular polyhedra, each having all its edges, faces and vertices equivalent. Second 58 59 to the Platonic solids in regularity come the Archimedean polyhedra, in which all vertices (but 60 not faces or edges) are equivalent. However, these two families provide us with only a limited number of ideal shapes and we should have at hand other sets of less regular polyhedra. For 3 http://mc.manuscriptcentral.com/poc Journal of Physical Organic Chemistry Page 4 of 20 1 2 3 4 instance, the prisms, whose ideal shapes are conventionally those with all edges of the same 5 length. As a result, all the faces of the ideal prisms are regular polygons. If more reference 6 [11] 7 shapes are needed we can make recourse to the 92 Johnson polyhedra, defined as those that 8 have as faces only regular polygons with edges of the same length, excluding the Platonic, 9 10 Archimedean, prismatic and antiprismatic polyhedra. Yet another important family of 11 polyhedra is that of the Fullerenes. Even if the name comes from C , Buckminsterfullerene, 12 60 13 that has the shape of the Archimedean truncated icosahedron, it refers to all those polyhedra 14 with twelve pentagonal and any number of hexagonal faces,[12] in which all vertices are three- 15 16 connected. 17 18 19 We note that the Platonic and Archimedean polyhedra belong to one of three high- 20 symmetry point groupFors, icosahedral Peer (Ih), octahedral Review (Oh) or tetrahedral (Td), or to their 21 [13] 22 rotational subgroups. In principle, icosahedral and octahedral symmetries are incompatible, 23 since the former features five-fold rotational axes, while the latter has four-fold axes instead.